Decide which is the correct factored form of the polynomial [tex]$6 y^2+41 y-56$[/tex]. Choose the correct factored form.
A. [tex]$(2 y+8)(3 y-7)$[/tex]
Answer (2)
The given factored form ( 2 y + 8 ) ( 3 y − 7 ) does not equal the polynomial 6 y 2 + 41 y − 56 . The correct factored form is ( 6 y − 7 ) ( y + 8 ) . Option A is therefore incorrect.
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Expand the given factored form ( 2 y + 8 ) ( 3 y − 7 ) and show that it does not equal 6 y 2 + 41 y − 56 .
Find two numbers that multiply to 6 × − 56 = − 336 and add up to 41 , which are 48 and − 7 .
Rewrite the middle term as 41 y = 48 y − 7 y and factor by grouping: 6 y 2 + 48 y − 7 y − 56 = 6 y ( y + 8 ) − 7 ( y + 8 ) .
The correct factored form is ( 6 y − 7 ) ( y + 8 ) .
Explanation
Checking the given option We are given the quadratic expression 6 y 2 + 41 y − 56 and asked to find its correct factored form. We are given option A: ( 2 y + 8 ) ( 3 y − 7 ) . Let's first expand this factored form to see if it matches the given quadratic expression.
Expanding the factored form Expanding ( 2 y + 8 ) ( 3 y − 7 ) , we get: ( 2 y + 8 ) ( 3 y − 7 ) = 2 y ( 3 y ) + 2 y ( − 7 ) + 8 ( 3 y ) + 8 ( − 7 ) = 6 y 2 − 14 y + 24 y − 56 = 6 y 2 + 10 y − 56.
Comparing with the original expression Comparing the expanded form 6 y 2 + 10 y − 56 with the original quadratic expression 6 y 2 + 41 y − 56 , we see that they are not the same. Therefore, option A is incorrect.
Finding the correct factors Now, let's find the correct factored form of 6 y 2 + 41 y − 56 . We are looking for two numbers that multiply to 6 × − 56 = − 336 and add up to 41 . We can list the factors of 336: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336 We need a pair of factors with a difference of 41. The pair 7 and 48 works, since 48 − 7 = 41 and 48 × 7 = 336 . So we can rewrite the middle term as 41 y = 48 y − 7 y .
Rewriting the expression Now we rewrite the quadratic expression as: 6 y 2 + 41 y − 56 = 6 y 2 + 48 y − 7 y − 56
Factoring by grouping We can factor by grouping: 6 y 2 + 48 y − 7 y − 56 = 6 y ( y + 8 ) − 7 ( y + 8 ) = ( 6 y − 7 ) ( y + 8 )
The correct factored form Therefore, the correct factored form of 6 y 2 + 41 y − 56 is ( 6 y − 7 ) ( y + 8 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and calculate stress and strain. In business, factoring can be used to optimize costs and maximize profits. Imagine you are designing a rectangular garden with an area represented by the quadratic expression 6 y 2 + 41 y − 56 . Factoring this expression allows you to determine the possible dimensions (length and width) of the garden, which in this case are ( 6 y − 7 ) and ( y + 8 ) . This helps in planning the layout and optimizing the use of space.
